From the relations (26), (28) and (27), (29) it can be seen that the set of operators
?
), Ai (q, q ) and Eµ (p, p ), Bµ? (q, q ) also form a continuous Lie algeb-
? i
Eµ (p, p µ?
ra.
74 W.I. Fushchych

For consideration of the continuous Lie algebras, we may introduce, by analogy to
the classical Lie algebra theory, the concepts of the universal enveloping Lie algebra,
the center, Casimir operators, etc. It is clear that all these concepts require refinement
from the mathematical point of view since, so far as we know, such Lie algebras are
not considered in the mathematical literature. As regards the problem of classification
and formulation of all irreducible representations of the algebra Sp2N (k, k ), it leads,
as can be seen from relations (16) and (17), to the problem of the description of all
unitary non-equivalent commutation relations (18). This last problem, as is known,
has not been solved up to the present time.
?
With the operators Eµ (p, p ), apparently, one cannot directly associate certain
physical quantities (energy, momentum, angular momentum, etc.). However, the in-
tegral operators derived from these operators, i.e., operators of the type

also form continuous Lie algebras.
3. In [7] it was shown that the set of infinitesimal operators of homogeneous
Lorentz group O(3, 1) and operators Lµ , entering into the relativistic equation

?
+ ? ?(x0 , x) = 0, (31)
Lµ µ = 0, 1, 2, 3
?xµ

form a Lie algebra, which is an isomorphous set of infinitesimal operators of the de Si-
tter group O(4, 1). The function ?(x0 , x) for a Lorentz transformation is transformed
n
li ,li
?Ri0 1 , where l0 , l1 are pairs of numbers
ii
according to the representation R =
i=1
to which are given the irreducible representations of O(3, 1). Since the generators of
group O(3, 1) and operators Lµ transform one solution of Eq.(31) to another solution,
it is clear that in all solution sets of (31) there are realized irreducible representations
of group O(4, 1). Since ?(x0 , x) pertains to a apace which is a linear sum of spaces in
which is realized the irreducible representation O(3, 1), then, obviously, the spectrum
of the Casimir operators,
1 1
K1 = ? Mµ? Mµ? , K2 = ? ?µ??? Mµ? M?? , µ, ?, ?, ? = 0, 1, 2, 3
2 4
in this space will be discrete.
On the basis of the above it is natural to propose the following problem: to
formulate an equation for the wave function ? which would be invariant relative to
A relativistically invariant mass operator 75

the Poincar? group and in all sets of solutions (solution space) of this equation of the
e
spectrum of Casimir operators,
1
P 2 = Pµ Pµ , W 2 = W? W? , (32)
W? = ????? P? M??
2
would be discrete.
For the solution of this problem we will use one of the results of Foldy [8]. In [8]
it was shown that with each irreducible unitary representation of the Poincar? group
e
with mass m and spin s there can be associated a Schr?dinger equation
o

??(x0 , x)
(33)
H?(x0 , x) = i ,
?t
where H = (P 2 + m2 )1/2 and ?(x0 , x) is the (2s + 1)-component wave function,
quadratically integrable over the space variables. The question of the uniqueness of
such correspondence (i.e., the question of possible existence of another equation which
would also express the free motion of a relativistic particle with mass m and spin s)
is left open in [8].
The single ambiguity, which apparently arises from the establishment of this
correspondence, is tied to the extraction of the square root of the operator P 2 + m2 .
Actually there is no such ambiguity, since the operator P 2 + m2 is positive, and by
virtue of theorems [10] the square root of a positive self-adjoint operator is uniquely
determined. This is proof in itself that the stated correspondence is isomorphic.
If the Hamiltonian in Eq.(34) is expressed in the form

then be seen that (34) is a natural generalization of the relativistic Eq.(33) (in which
the constant value m2 is replaced by the operator M 2 ) in the case where the particle
can take on various mass states.
In this manner, every relativistic equation expressing a free particle of mass m
and spin s is unitarily equivalent to Eq.(33) (H > 0).
Since the Casimir operators P 2 and W 2 enter the theory on equal terms, then we
may use the operator W 2 to obtain the equation of motion of a free particle. In this
case, the equation which, generally speaking, is unitarily equivalent to Eq.(33), has
the form

W 2 + m2 s(s + 1)X(x, t) = W0 X(x, t), (33 )

i.e., between X and ?, there exists the coupling X = V ?, where V is the isometric
operator.
Establishment of isomorphism between the Schr?dinger equations and the irredu-
o
cible unitary representations of the Poincar? group permits the writing of the equation
e
which would have the above state properties. This equation has the form

The plus sign means that the sign value of the supplementary Casimir operator (the
sign of the energy) for the Poincar? group [9] for these solutions is equal to +1.
e
The Schr?dinger equation which would also be invariant under time reflection has
o
the form
?X
(35)
H X=i ,
?t

H0 ?+
where H = ,X= .
0H ??
In conclusion, let us note that, in agreement with the theorems of O’Raifeltai-
gh [13] in the space of the solutions of Eq.(34) one cannot realize an irreducible
representation of a finite dimensional Lie algebra which would contain the Poincar? e
algebra as a subalgebra.
If Eqs. (31) and (33) are considered equivalent (the unitary equivalence is con-
structed only for equations describing particles with spin 1/2), then the formula for
? (x0 , x), expressing motion of a particle which may be in various mass states, has
the same formal appearance as the equations for elementary particles. However, the
quantity ? is then not a constant but a variable, taking on the following values:

? = ±m1 ?1 , ±m2 ?2 , ±m3 ?3 , . . . , (36)

where m2 = p2 ? p2 and ?i is some real nonzero eigenvalue of the operator L0 .
0
i
In [15] it is shown that only for such values of ? do Eqs.(31) have nonzero plane
wave solutions.
The relation (36) can be written in the form of the mass formula:

M = ?L?1 . (36 )
0

The operators which transform solutions of Eq.(31) with fixed mass to solutions
which have a different mass are constructed from creation and annihilation operators
by an analogous method (as in Sec. 1).
Note 2. Equation (31), as was shown in [11], excluding the Dirac equation, cannot be
reduced by the unitary representations of the Foldy–Woythysen type to a Schr?dinger
o
equation. Consequently, the function ?(x0 , x), strictly speaking, is not a wave function
of a particle with fixed mass m and spin s.
The construction of a non-trivial theory of interaction based on Eq.(35), i.e., the
introduction of potential in (35), by excluding those theories which with the help
of unitary representations reduce to free particles (or as is generally stated, to the
theory of free quasiparticles) [3], meets with difficulties in practice [14].
From the previous considerations, with every elementary particle there is asso-
ciated a space Ri , in which is realized an irreducible representation of algebra P .
A relativistically invariant mass operator 77

A particle which can be found in various excited states is associated with space R
which is a linear sum of the spaces Ri . The inadequacy of such an approach lies
in the fact that all elementary particles are considered as stable, and consequently
possessing definite mass. Actually, a definite mass to these resonances cannot be